1.3: The Arithmetic and Composition of Functions
- Refresher Topics
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Simplifying Expressions
- Multiplying and Dividing Rational Expressions
- Adding and Subtracting Rational Expressions
- Simplifying Compound Rational Expressions
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Factoring Techniques
- Factoring Difference of Squares
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Solving Equations
- Solving Rational Equations (involving a single rational expression)
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Simplifying Expressions
In this section, you will:
- Perform arithmetic on functions
- Combine functions using algebraic operations.
- Create a new function by composition of functions.
- Evaluate composite functions.
- Find the domain of a composite function.
- Decompose a composite function into its component functions.
- Compute the Difference Quotient for a given function.
Suppose we want to calculate how much it costs to heat a house on a particular day of the year. This cost will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The function \( C ( T ) \) gives the cost \( C \) of heating a house for a given average daily temperature in \( T \) degrees Celsius. The function \( T ( d ) \) gives the average daily temperature on day \( d \) of the year. For any given day, \( \text{Cost\} = C ( T ( d ) ) \) means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature \( T ( d ) \). For example, we could evaluate \( T ( 5 ) \) to determine the average daily temperature on the 5 th day of the year. Then, we could evaluate the cost function at that temperature. We would write \( C ( T ( 5 ) ) \).
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
Combining Functions Using Algebraic Operations
Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wife's separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year's incomes and then collecting all the data in a new column. If \( w(y) \) is the wife's income and \( h ( y ) \) is the husband's income in year \( y \), and we want \( T \) to represent the total income, then we can define a new function.\[ T ( y ) = h ( y ) + w ( y ). \nonumber \]If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write\[ T = h + w. \nonumber \]Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of Algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions \( f(x) \) and \( g(x) \) with real number outputs, we define new functions \( f + g \), \(f − g\), \(f g\), and \( \frac{f}{g} \) by the relations\[ \begin{array}{rcl}
( f + g ) ( x ) & = & f ( x ) + g ( x ) \\[6pt]
( f - g ) ( x ) & = & f ( x ) - g ( x ) \\[6pt]
( f g ) ( x ) & = & f ( x ) g ( x ) \\[6pt]
\left( \dfrac{f}{g} \right) ( x ) & = & \dfrac{f ( x )}{g ( x )} \\[6pt]
\end{array} \nonumber \]
Find and simplify the functions \( ( g − f ) ( x ) \) and \( \left( \frac{g}{f}\right) ( x ) \), given \( f ( x ) = x − 1 \) and \( g ( x ) = x^2 − 1 \). Are they the same function?
- Solution
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Begin by writing the general form, and then substitute the given functions. For \( (g - f)(x) \), we have\[ \begin{array}{rclcl}
(g - f)(x) & = & g(x) - f(x) & & \\[6pt]
& = & x^2 - 1 - (x - 1) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & x^2 - 1 - x + 1 & \quad & \left( \text{distributing} \right) \\[6pt]
& = & x^2 - x & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & x(x - 1) & \quad & \left( \text{factoring out the GCF} \right) \\[6pt]
\end{array} \nonumber \]For \( \left( \frac{g}{f} \right)(x) \), we have\[ \begin{array}{rclcl}
\left( \dfrac{g}{f} \right)(x) & = & \dfrac{g(x)}{f(x)} & & \\[6pt]
& = & \dfrac{x^2 - 1}{x - 1} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{(x - 1)(x + 1)}{x - 1} & \quad & \left( \text{factoring} \right) \\[6pt]
& = & \dfrac{\cancelto{1}{(x - 1)}(x + 1)}{\cancelto{1}{(x - 1)}} & \quad & \left( \text{canceling like terms and }\textbf{assuming }x \neq 1 \right) \\[6pt]
& = & x + 1 & \quad & \left( \text{where }x \neq 1 \right) \\[6pt]
\end{array} \nonumber \]Therefore, the functions are not the same.
In Example \( \PageIndex{ 1 } \), for \( \left( \frac{g}{f} \right) ( x ) \), the condition \( x \neq 1 \) is necessary because when \( x = 1\), the denominator is equal to 0, which makes the function undefined.
Find and simplify the functions \( ( f g ) ( x ) \) and \( ( f − g ) ( x ) \) given\[ f ( x ) = x − 1 \text{ and } g ( x ) = x^2 − 1.\nonumber \]Are they the same function?
Focusing on Calculus - Difference Quotients
Now that we know how to perform arithmetic with functions, it's time to turn our attention to the primary building block of a huge concept in Calculus. This primary building block is called the difference quotient .
Given a function \(f\), the difference quotient of \(f\) is the expression\[\dfrac{f(x+h)-f(x)}{h}. \label{dq} \]
You might see this written online and in other texts as\[ \dfrac{f(a + h) - f(a)}{h}, \nonumber \]but fear not - this is the same expression (just with \( a \) instead of \( x \)).
We will revisit this concept in more detail in a future section, but for now we use it as a way to practice function notation and function arithmetic. For reasons that will become clear in Calculus, "simplifying" a difference quotient means rewriting it in a form where the \(h\) in Equation \( \ref{dq} \) cancels from the denominator. Once that happens, we consider our work to be done.
Find and simplify the difference quotient for the following function.\[f(x)=x^{2}-x-2\nonumber \]
- Solution
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To find \(f(x+h)\), we replace every occurrence of \(x\) in the formula \(f(x)=x^{2}-x-2\) with the quantity \((x + h)\) to get\[ \begin{array}{rclcl}
f(x+h) & = & (x+h)^{2}-(x+h)-2 & \quad & \left( \text{substituting} \right)\\[6pt]
& = & x^{2}+2 x h+h^{2}-x-h-2 & \quad & \left( \text{distributing} \right) \\[6pt]
\end{array} \nonumber \]So the difference quotient is\[ \begin{array}{rclcl}
\dfrac{f(x+h)-f(x)}{h} & = & \dfrac{\left(x^{2}+2 x h+h^{2}-x-h-2\right)-\left(x^{2}-x-2\right)}{h} & & \\[6pt]
& = & \dfrac{x^{2}+2 x h+h^{2}-x-h-2-x^{2}+x+2}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{2 x h+h^{2}-h}{h} & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & \dfrac{h(2 x+h-1)}{h} & \quad & \left( \text{factoring out the GCF} \right)\\[6pt]
& = & \dfrac{\cancel{h}(2 x+h-1)}{\cancel{h}} & \quad & \left( \text{canceling like factors} \right)\\[6pt]
& = & 2x + h - 1. \\[6pt]
\end{array} \nonumber \]We know we are done once the original \( h \) in the denominator is canceled out.
A word of caution before we continue: watch your negatives! Notice that at some point in the solution of Example \( \PageIndex{ 2 } \) we had the expression \( \left(x^{2}+2 x h+h^{2}-x-h-2\right)-\left(x^{2}-x-2\right) \) in the numerator. Those parentheses on the second group are critically important and necessary. We are subtracting
the entire function
\( f(x) \). As such, we needed \( f(x + h) - f(x) = f(x + h) - \left( x^2 - x - 2 \right) \). If we didn't have those parentheses, we would (incorrectly) have written \( f(x + h) - f(x) = f(x + h) - x^2 - x - 2 \), which I hope you would agree is not the same thing.
Let's state that one more time for clarity -
the biggest mistake made on a difference quotient problem is not using grouping symbols properly!
You have been warned.
Find and simplify the difference quotient for \( g(x) = 1 - 3x \).
Create a Function by Composition of Functions
Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions . The resulting function is known as a composite function . We represent this combination by the following notation:\[ ( f \circ g ) ( x ) = f ( g ( x ) ).\nonumber \]We read the left-hand side as "\( f \) composed with \( g \) at \( x \)," and the right-hand side as "\( f \) of \( g \) of \( x \)." The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol \( \circ \) is called the composition operator . We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value.
Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication take two numbers and give a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases \( f ( g ( x ) ) \neq f ( x ) g ( x )\).
It is also important to understand the Order of Operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function \( g \) takes the input \( x \) first and yields an output \( g ( x ) \). Then the function \( f \) takes \( g(x) \) as an input and yields an output \( f ( g ( x ) ) \).
In general, \( f \circ g \) and \( g \circ f \) are different functions. In other words, in many cases \( f ( g ( x ) ) \neq g ( f ( x ) ) \) for all \( x \). We will also see that sometimes two functions can be composed only in one specific order.
For example, if \( f ( x ) = x^2 \) and \( g ( x ) = x + 2 \), then\[ \begin{array}{rclcl}
f(g(x)) & = & f(x + 2) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & (x + 2)^2 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & x^2 + 4x + 4 & \quad & \left( \text{distributing} \right) \\[6pt]
\end{array} \nonumber \]but\[ \begin{array}{rclcl}
g(f(x)) & = & g(x^2) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & x^2 + 2 & \quad & \left( \text{substituting} \right) \\[6pt]
\end{array} \nonumber \]These expressions are not equal for all values of \( x \), so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value \( x = −\frac{1}{2} \).
Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.
When the output of one function is used as the input of another, we call the entire operation a composition of functions . For any input \( x \) and functions \( f \) and \( g \), this action defines a composite function , which we write as \( f \circ g \) such that\[ ( f \circ g ) ( x ) = f ( g ( x ) ). \nonumber \]The domain of the composite function \( f \circ g \) is all \( x \) such that \( x \) is in the domain of \( g \) and \( g(x) \) is in the domain of \( f \).
It is important to realize that the product of functions \( f g \) is not the same as the function composition \( f ( g ( x ) ) \), because, in general, \( f ( x ) g ( x ) \neq f ( g ( x ) ) \).
Using the functions provided, find \( f ( g ( x ) ) \) and \( g ( f ( x ) ) \). Determine whether the composition of the functions is commutative .\[ f ( x ) = 2 x + 1, \quad g ( x ) = 3 − x. \nonumber \]
- Solution
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Let's begin by substituting \( g(x) \) into \( f ( x ) \).\[ \begin{array}{rclcl}
f(g(x)) & = & f(3 - x) + 1 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 2(3 - x) + 1 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 6 - 2x + 1 & \quad & \left( \text{distributing} \right) \\[6pt]
& = & 7 - 2x & \quad & \left( \text{combining like terms} \right) \\[6pt]
\end{array} \nonumber \]Now we can substitute \( f(x) \) into \( g ( x ) \).\[ \begin{array}{rclcl}
g(f(x)) & = & g(2x + 1) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 3 - (2x + 1) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 3 - 2x - 1 & \quad & \left( \text{distributing} \right) \\[6pt]
& = & 2 - 2x - 1 & \quad & \left( \text{combining like terms} \right) \\[6pt]
\end{array} \nonumber \]We find that \( g ( f ( x ) ) \neq f ( g ( x ) ) \), so the operation of function composition is not commutative.
The function \( c ( s ) \) gives the number of calories burned completing \( s \) sit-ups, and \( s ( t ) \) gives the number of sit-ups a person can complete in \( T \) minutes. Interpret \( c ( s ( 3 ) ) \).
- Solution
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The inside expression in the composition is \( s ( 3 ) \). Because the input to the \( s \)-function is time, \( t = 3 \) represents 3 minutes, and \( s ( 3 ) \) is the number of sit-ups completed in 3 minutes.
Using \( s ( 3 ) \) as the input to the function \( c ( s ) \) gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
Suppose \( f(x) \) gives miles that can be driven in \( x \) hours and \( g(y) \) gives the gallons of gas used in driving \( y \) miles. Which of these expressions is meaningful: \( f ( g ( y ) ) \) or \( g ( f ( x ) ) \)?
- Solution
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The function \( y = f ( x ) \) is a function whose output is the number of miles driven corresponding to the number of hours driven.\[ \text{number of miles} = f ( \text{number of hours} ). \nonumber \]The function \( g(y) \) is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:\[ \text{number of gallons} = g ( \text{number of miles} ). \nonumber \]The expression \( g(y) \) takes miles as the input and a number of gallons as the output. The function \( f(x) \) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression \( f ( g ( y ) ) \) is meaningless.
The expression \( f(x) \) takes hours as input and a number of miles driven as the output. The function \( g(y) \) requires a number of miles as the input. Using \( f(x) \) (miles driven) as an input value for \( g ( y ) \), where gallons of gas depends on miles driven, does make sense. The expression \( g ( f ( x ) ) \) makes sense, and will yield the number of gallons of gas used, \( g \), driving a certain number of miles, \( f ( x ) \), in \( x \) hours.
Are there any situations where \( f ( g ( y ) ) \) and \( g ( f ( x ) ) \) would both be meaningful or useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.
The gravitational force on a planet a distance \( r \) from the sun is given by the function \( G ( r ) \). The acceleration of a planet subjected to any force \( F \) is given by the function \( a ( F ) \). Form a meaningful composition of these two functions, and explain what it means.
Evaluating Composite Functions
Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.
Evaluating Composite Functions Using Tables
When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.
Using the following table, evaluate \( f ( g ( 3 ) ) \) and \( g ( f ( 3 ) ) \).
| \( x \) | \( f(x) \) | \( g(x) \) |
|---|---|---|
| 1 | 6 | 3 |
| 2 | 8 | 5 |
| 3 | 3 | 2 |
| 4 | 1 | 7 |
- Solution
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To evaluate \( f ( g ( 3 ) ) \), we start from the inside with the input value 3. We then evaluate the inside expression \( g ( 3 ) \) using the table that defines the function \( g \): \( g ( 3 ) = 2 \). We can then use that result as the input to the function \( f \), so \( g ( 3 ) \) is replaced by 2 and we get \( f ( 2 ) \). Then, using the table that defines the function \( f \), we find that \( f ( 2 ) = 8 \).\[ \begin{array}{rrcl}
& g(3) & = & 2 \\[6pt]
\implies & f(g(3)) & = & f(2) = 8 \\[6pt]
\end{array} \nonumber \]To evaluate \( g ( f ( 3 ) ) \), we first evaluate the inside expression \( f ( 3 ) \) using the first table: \( f ( 3 ) = 3 \). Then, using the table for \( g \), we can evaluate\[ g(f(3)) = g(3) = 2. \nonumber \]The following table shows the composite functions \( f \circ g \) and \( g \circ f \) as tables.\( x \) \( g(x) \) \( f ( g ( x ) ) \) \( f(x) \) \( g ( f ( x ) ) \) 3 2 8 3 2
Use the table from Example \( \PageIndex{ 6 } \) to evaluate \( f ( g ( 1 ) ) \) and \( g ( f ( 4 ) ) \).
Evaluating Composite Functions Using Graphs
When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the \( x \)- and \( y \)-axes of the graphs.
Use the following figure to evaluate \( f ( g ( 1 ) ) \).
- Solution
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To evaluate \( f ( g ( 1 ) ) \), we start with the inside evaluation.
We evaluate \( g ( 1 ) \) using the graph of \( g ( x ) \), finding the input of 1 on the \( x \)-axis and finding the output value of the graph at that input. Here, \( g ( 1 ) = 3 \). We use this value as the input to the function \( f \).\[ f ( g ( 1 ) ) = f ( 3 )\nonumber \]We can then evaluate the composite function by looking to the graph of \( f ( x ) \), finding the input of 3 on the \( x \)-axis and reading the output value of the graph at this input. Here, \( f ( 3 ) = 6 \), so \( f ( g ( 1 ) ) = 6 \). - Analysis
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The following figure shows how we can mark the graphs with arrows to trace the path from the input value to the output value.
Use the graph from Example \( \PageIndex{ 7 } \) to evaluate \( g ( f ( 2 ) ) \).
Evaluating Composite Functions Using Formulas
When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition \( f ( g ( x ) ) \). To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like \( f ( t ) = t^2 − t \), we substitute the value inside the parentheses into the formula wherever we see the input variable.
Given \( f ( t ) = t^2 − t \) and \( h ( x ) = 3 x + 2\), evaluate \( f ( h ( 1 ) ) \).
- Solution
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Because the inside expression is \( h ( 1 ) \), we start by evaluating \( h ( x ) \) at 1.\[ \begin{array}{rclcl}
h(1) & = & 3(1) + 2 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 5 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array} \nonumber \]Then \( f ( h ( 1 ) ) = f ( 5 ) \), so we evaluate \( f ( t ) \) at an input of 5.\[ \begin{array}{rclcl}
f(h(1)) & = & f(5) & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 5^2 - 5 & \quad & \left( \text{substituting} \right) \\[6pt]
& = & 25 - 5 & \quad & \left( \text{squaring} \right) \\[6pt]
& = & 20 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array} \nonumber \] - Analysis
- It makes no difference what the input variables \( T \) and \( x \) were called in this problem because we evaluated for specific numerical values.
Given \( f ( t ) = t^2 − t \) and \( h ( x ) = 3 x + 2 \), evaluate
- \( h ( f ( 2 ) ) \)
- \( h ( f ( − 2 ) ) \)
Finding the Domain of a Composite Function
As we discussed previously, the domain of a composite function such as \( f \circ g \) is dependent on the domain of \( g \) and the domain of \( f \). It is important to know when we can apply a composite function and when we cannot. That is, it is important to know the domain of a function such as \( f \circ g \).
Let us assume we know the domains of the functions \( f \) and \( g \) separately. If we write the composite function for an input \( x \) as \( f ( g ( x ) ) \), we can see right away that \( x \) must be a member of the domain of \( g \) in order for the expression to be meaningful. Otherwise, we cannot complete the inner function evaluation. However, we also see that \( g(x) \) must be a member of the domain of \( f \), otherwise the second function evaluation in \( f ( g ( x ) ) \) cannot be completed and the expression is still undefined. Thus, the domain of \( f \circ g \) consists of only those inputs in the domain of \( g \) that produce outputs from \( g \) belonging to the domain of \( f \). Note that the domain of \( f \) composed with \( g \) is the set of all \( x \) such that \( x \) is in the domain of \( g \) and \( g(x) \) is in the domain of \( f \).
The domain of a composite function \( f ( g ( x ) ) \) is the set of those inputs \( x \) in the domain of \( g \) for which \( g(x) \) is in the domain of \( f \).
To find the domain of the composite function \( f(g(x)) \), we start by finding the domain of \( g \). Those values of \( x \) for which \( g \) is defined will be the largest possible domain of the composition; however, this set might get restricted once we take the function \( f \) into consideration. That said, note the domain of \( f \). Our goal is to find the elements, \( x \), in the domain of \( g \) for which \( g(x) \) is in the domain of \( f \). That is, exclude those inputs \( x \) from the domain of \( g \) for which \( g(x) \) is not in the domain of \( f \). The resulting set is the domain of \( f \circ g \).
Find the domain of \( ( f \circ g ) ( x ) \) where \( f ( x ) = \frac{5}{x − 1} \) and \( g ( x ) = \frac{4}{3 x − 2} \).
- Solution
-
The domain of \( g(x) \) consists of all real numbers except \( x = \frac{2}{3} \), since that input value would cause us to divide by 0. Likewise, the domain of \( f \) consists of all real numbers except 1. So we need to exclude from the domain of \( g(x) \) that value of \( x \) for which \( g ( x ) = 1 \).\[ \begin{array}{rrclcl}
& g(x) & = & 1 & & \\[6pt]
\implies & \dfrac{4}{3x - 2} & = & 1 & \quad & \left( \text{substituting} \right) \\[6pt]
\implies & 4 & = & 1(3x - 2) & \quad & \left( \text{multiplying both sides by the LCD} \right) \\[6pt]
\implies & 4 & = & 3x - 2 & \quad & \left( \text{distributing} \right) \\[6pt]
\implies & 6 & = & 3x & \quad & \left( \text{adding }2\text{ to both sides} \right) \\[6pt]
\implies & 2 & = & x & \quad & \left( \text{dividing both sides by }3 \right) \\[6pt]
\end{array} \nonumber \]So the domain of \( f \circ g \) is the set of all real numbers except \( \frac{2}{3} \) and \( 2 \). This means that \( x \neq \frac{2}{3} \) o \( x \neq 2 \).We can write this in interval notation as\[ \left( − \infty , \dfrac{2}{3} \right) \cup \left( \dfrac{2}{3} , 2 \right) \cup ( 2 , \infty ). \nonumber \]
Find the domain of \( ( f \circ g ) ( x ) \) where \( f ( x ) = \sqrt{ x + 2 } \) and \( g ( x ) = \sqrt{ 3 − x } \).
- Solution
-
Because we cannot take the square root of a negative number, the domain of \( g \) is \( ( − \infty , 3 ] \). Now we check the domain of the composite function\[ ( f \circ g ) ( x ) = \sqrt{\sqrt{ 3 − x } + 2 }. \nonumber \]Since the radicand of a square root must be positive, \( \sqrt{ 3 − x } + 2 \geq 0 \). Hence,\[ \begin{array}{rrclcl}
& \sqrt{ 3 − x } + 2 & \geq & 0 & & \\[6pt]
\implies & \sqrt{ 3 − x } & \geq & -2 & \quad & \left( \text{subtracting }2\text{ from both sides} \right) \\[6pt]
\end{array} \nonumber \]Since square roots never return negative numbers, the inequality \( \sqrt{3 - x} \geq -2 \) is always true. Thus, our only restriction on this composition is the original restriction from the domain of \( g \) - namely that the values of \( x \) must be in the interval \( ( − \infty , 3 ] \). This gives a domain of \( ( - \infty , 3 ] \). - Analysis
- This example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in finding the domain of a composite function. It also shows that the domain of \( f \circ g \) can contain values that are not in the domain of \( f \), though they must be in the domain of \( g \).
If we change the function \( f \) very slightly in Example \( \PageIndex{ 10 } \) to \( f(x) = \sqrt{x - 2} \), then the composition, \( (f \circ g)(x) \) becomes\[ (f \circ g)(x) = \sqrt{\sqrt{3 - x} - 2}. \nonumber \]Again, since the radicand is never negative, we know\[ \begin{array}{rrclcl}
& \sqrt{ 3 − x } - 2 & \geq & 0 & & \\[6pt]
\implies & \sqrt{ 3 − x } & \geq & 2 & \quad & \left( \textbf{adding }2\text{ to both sides} \right) \\[6pt]
\end{array} \nonumber \]This makes the problem of finding the domain a more complex. If \( \sqrt{3 - x} \geq 2 \), then\[ \left( \sqrt{3 - x} \right)^2 \geq 2^2 \implies 3 - x \geq 4 \implies 3 \geq 4 + x \implies -1 \geq x. \nonumber \]Hence, we would require \( x \leq 3 \) (from the domain of \( g \)) and, simultaneously, \( -1 \geq x \). The common intersection of these two domain restrictions is \( x \leq -1 \). Thus, the domain (in this adjusted case) would be \( ( -\infty,-1] \).
Find the domain of \( ( f \circ g ) ( x ) \) where \( f ( x ) = \frac{1}{x − 2} \) and \(g ( x ) = \sqrt{ x + 4 } \).
Decomposing a Composite Function into its Component Functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.
Write \( f ( x ) = \sqrt{ 5 − x^2 } \) as the composition of two functions.
- Solution
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We are looking for two functions, \( g \) and \( h \), so \( f ( x ) = g ( h ( x ) ) \). To do this, we look for a function inside a function in the formula for \( f ( x ) \). As one possibility, we might notice that the expression \( 5 − x^2 \) is the inside of the square root. We could then decompose the function as\[ h ( x ) = 5 − x^2 \text{ and } g ( x ) = \sqrt{ x }. \nonumber \]We can check our answer by recomposing the functions.\[ g ( h ( x ) ) = g ( 5 − x^2 ) = \sqrt{ 5 − x^2 }. \nonumber \]
Write \( f ( x ) = \frac{4}{3 − \sqrt{ 4 + x^2 }} \) as the composition of two functions.
Access these online resources for additional instruction and practice with composite functions.


10.1: Polymerization - Making Big Ones Out of Little Ones
- Define the terms monomer and polymer .
- Know the different types of natural polymers.
A polymer is a large molecule, or macromolecule, composed of many repeated subunits.The term "polymer" derives from the Greek word polus ( meaning "many, much") and m eros (meaning "part"), and refers to a molecule whose structure is composed of multiple repeating units, from which originates a characteristic of high relative molecular mass and attendant properties. As shown schematically in Figure \(\PageIndex{1}\).
Due to their broad range of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass relative to small molecule compounds produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures rather than crystals. The terms polymer and resin are often synonymous with plastic.
Natural Polymers
Some very important biological materials are polymers. Of the three major food groups, polymers are represented in two: proteins and carbohydrates. Proteins are polymers of amino acids, which are monomers that have an amine functional
group and a carboxylic acid functional group. Proteins play a crucial role in living organisms.Linking hundreds of glucose molecules together make
s a relatively common material known as starch :
Starch is an important source of energy in the human diet. Note how individual glucose units are joined together. They can also be joined together in another way, like this:
This polymer is known as cellulose . Cellulose is a major component in the cell walls of plants. Curiously, despite the similarity in the building blocks, some animals (such as humans) cannot digest cellulose; those animals that can digest cellulose typically rely on symbiotic bacteria in the digestive tract for the actual digestion. Animals do not have the proper enzymes to break apart the glucose units in cellulose, so it passes through the digestive tract and is considered dietary fiber .
Deoxyribonucleic acid ( DNA ) and ribonucleic acid ( RNA ) are also polymers, composed of long, three-part chains consisting of phosphate groups, sugars with 5 C atoms (ribose or deoxyribose), and N-containing rings referred to as bases. Each combination of the three parts is called a nucleotide; DNA and RNA are essentially polymers of nucleotides that have rather complicated but intriguing structures ( F igure \(\PageIndex{2}\) - Nucleotides). DNA is the fundamental material in chromosomes and is directly responsible for heredity, while RNA is an essential substance in protein synthesis.
The DNA in our cells is a polymer of nucleotides, each of which is composed of a phosphate group, a sugar, and a N-containing base
The above mentioned biopolymers (polymers produced by living organisms) are discussed further in Chapter 16.
Celluloid: Billiard Balls
Celluloids are a class of compounds created from nitrocellulose (partially nitrated cellulose) and camphor, with added dyes and other agents. Generally considered the first thermoplastic, it was first created as Parkesinein (by Alexander Parkes of Birmingham England) in 1856 and as Xylonite in 1869. In the 1860s, an American, John Wesley Hyatt, acquired Parkes's patent and began experimenting with cellulose nitrate with the intention of manufacturing billiard balls, which until that time were made from ivory. In the 1870s the modified plastic was registered as "celluloid".
The main use was in movie and photography film industries, which used only celluloid film stock prior to the adoption of acetate safety film in the 1950s. Celluloid is highly flammable, difficult and expensive to produce and no longer widely used; its most common uses today are in table tennis balls, musical instruments, and guitar picks.
Bakelite (sometimes spelled Baekelite ) or polyoxybenzylmethylenglycolanhydride was the first plastic made from synthetic components. It is a thermosetting phenol formaldehyde resin, formed from a condensation reaction of phenol with formaldehyde. It was developed by the Belgian-American chemist Leo Baekeland in Yonkers, New York, in 1907.
Bakelite was patented on December 7, 1909. The creation of a synthetic plastic was revolutionary for its electrical non conductivity and heat-resistant properties in electrical insulators, radio and telephone casings and such diverse products as kitchenware, jewelry, pipe stems, children's toys, and firearms.
Video \(\PageIndex{2}\) Polymers Crash Course
Summary
- Polymers are giant molecules that consist of long chains of units called monomers connected by covalent bonds.
- Polymerization is the process of linking monomers together to form a polymer.
- Plastic is the general term for polymers made from synthetic materials.
- Several important biological polymers include proteins, starch, cellulose, DNA and RNA.
Contributors and Attributions
- Joshua Halpern, Scott Sinex and Scott Johnson
- TextMap: Beginning Chemistry (Ball et al.)
- Wikipedia


5.4: For Profit or Not
As you learned earlier, a social entrepreneur is an innovator seeking social benefits. But the story doesn’t end there. Social entrepreneurs can be found in the for-profit and non-profit sectors. Non-profits have traditionally been tasked, along with government, with providing public benefit. But an increasing number of for-profit companies are also tackling social and environmental issues.
Non-profit doesn’t mean “no profit”
If your business serves a cause or does a public good, it can earn a profit and still be a non-profit organization. The US government rewards some organizations that perform services to the community by letting them operate tax-free. This is to recognize the fact that the government itself would have to spend more money to tackle social problems without the help of these groups. Many charitable groups rely solely on donations for funding, and pay no taxes on the money they take in. However, groups that want to fund their cause by selling items or providing services for a fee can make profits (i.e., “earned income”), benefit from this tax status, and call themselves non-profits, too. Non-profit groups with earned income strategies often refer to their money-making activities as “social ventures” and to their founders as “social entrepreneurs.”
Money to the mission
“Non-profit” is short for “not-for-profit,” meaning that money earned (through any means) is reinvested towards a mission. This contrasts with for-profit companies which divide income, after expenses, among owners or shareholders. In a good year, the owners of a for-profit company get richer, while a non-profit has more resources to devote to its cause. Non-profits are considered to be owned by the public, and cannot be used for the personal gain of individuals.
The most common type of charitable organization is a “501(c)3,” as identified by the IRS. Companies that meet this classification are exempt from state and federal income, sales, and property taxes. However, they must have a clearly defined mission from which they operate. All income must be generated in a way that relates to the mission, and after expenses, must be devoted to that mission. This prevents charities from using their tax-free status to compete with other merchants who have higher expenses due to taxes. 501(c)3 organizations cannot support political causes as part of their operation.
A non-profit must have a board of directors. As with for-profit companies, which maintain boards to insure that management acts according to shareholder interests, non-profit boards insure that management works in the best interest of the targeted community.
Profit and mission
You can incorporate socially beneficial elements into your company, but still run it as a for-profit. This allows you to use profits to motivate owners, financers, and employees (through stock options) and have market discipline while keeping a strong focus on mission-related results. You can consider yourself a for-profit, socially responsible business, or a for-profit social venture. If you want to focus on your ideals, articulate them in your corporate mission statement and find ways to measure progress beyond financial statements. See the Plan section for ideas.
Other considerations
Now that you know how the IRS defines for-profit and non-profit companies, consider other factors besides tax benefits to decide where your company fits. In their paper “Blurring Sector Boundaries: Serving Social Purposes through For-Profit Structures,” J. Gregory Dees and Beth B. Anderson at The Fuqua School of Business of Duke University consider the following:
- Public view of organization/mission
- Availability of financing
- Incentives for employees and owners
- Ability to stick with mission
- Market discipline
The non-profit sector has a reputation for being mission-oriented—having this status makes the organizational focus more believable to the public. Their inability to reward owners, employees, and financiers with financial returns tied to performance may restrict sources of money and people, but removes the temptation to sacrifice the cause for higher profits.
Economists believe that a well-functioning market is the best way to determine the value of an entity. For-profit companies issue stocks that are publicly traded, which enables their worth to be determined by the stock price. The stock price represents the investors’ appraisal of the future value of the company’s profits. Fluctuations send a signal to managers that the market thinks they’re making good or bad decisions. This feedback is termed market discipline. Arguably because investments in for-profit entities can be publicly traded, they must be more attentive to their owners and the signals in the market. Some see a disadvantage in the fact that non-profits cannot get these signals as efficiently from the stakeholders that “own” them.
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Board basics: Overview of a nonprofit board
From www.ascs.org
Starting a non-profit
Staff
Budget
Partnering
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11.2: Barriers to Entry: Reasons for Monopolies to Exist
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Resource Control
Control over a natural resource that is critical to the production of a final good is one source of monopoly power.
Learning Objectives
- Explain the relationship between resource control and monopolies
Control over natural resources that are critical to the production of a good is one source of monopoly power. Single ownership over a resource gives the owner of the resource the power to raise the market price of a good over marginal cost without losing customers to competitors. In other words, resource control allows the controller to charge economic rent. This is a classic outcome of imperfectly competitive markets.
A classic example of a monopoly based on resource control is De Beers. De Beers Consolidated Mines were founded in 1888 in South Africa as an amalgamation of a number of individual diamond mining operations. De Beers had a monopoly over the production of diamonds for most of the 20th century, and it used its dominant position to manipulate the international diamond market. It convinced independent producers to join its single channel monopoly. In instances when producers refused to join, De Beers flooded the market with diamonds similar to the ones they were producing. De Beers also purchased and stockpiled diamonds produced by other manufacturers in order to control prices through supply. The De Beers model changed at the turn of the 21st century, when diamond producers from Russia, Canada, and Australia started to distribute diamonds outside of the De Beers channel. The sale of diamonds also suffered from rising awareness about blood diamonds. De Beers’ market share fell from as high as 90 percent in the 1980s to less than 40 percent in 2012.
Diamonds : For most of the 20th century, De Beers had monopoly power over the world market for diamonds.
In practice, monopolies rarely arise because of control over natural resources. Economies are large, usually with multiple people owning resources. International trade is an additional source of competition for owners of natural resources.
Economies of Scale and Network Externalities
Economies of scale and network externalities discourage potential competitors from entering a market.
Learning Objectives
- Define Economies of Scale., Explain why economies of scale are desirable for monopolies
Economies of scale and network externalities are two types of barrier to entry. They discourage potential competitors from entering a market, and thus contribute to the monopolistic power of some firms.
Economies of scale are cost advantages that large firms obtain due to their size.They occur because the cost per unit of output decreases with increasing scale, as fixed costs are spread over more units of output. Economies of scale are also gained through bulk-buying of materials with long-term contracts, the increased specialization of managers, ability to obtain lower interest rates when borrowing from banks, access to a greater range of financial instruments, and spreading the cost of marketing over a greater range of output. Each of these factors contributes to reductions in the long-run average cost of production.
Economies of Scale : Large firms obtain economies of scale in part because fixed costs are spread over more units of output.
A natural monopoly arises as a result of economies of scale. For natural monopolies, the average total cost declines continually as output increases, giving the monopolist an overwhelming cost advantage over potential competitors. It becomes most efficient for production to be concentrated in a single firm.
Network externalities (also called network effects) occur when the value of a good or service increases as a result of many people using it. Because of network effects, certain goods or services that are adopted widely will appear to be much more attractive to new customers than competing goods or services. This is evident in online social networks. Social networks with the largest memberships are more attractive to new users, because new users know that their friends or colleagues are more likely to be on these networks. It is also evident with certain software programs. For example, most people use Microsoft word processing software. While other word processing programs may be available, an individual would risk running into compatibility problems when sending files to people or machines using the mainstream software. This makes it difficult for new companies to enter the market and to gain market share.
Government Action
here are two types of government-initiated monopoly: a government monopoly and a government-granted monopoly.
Learning objectives
- Discuss different types of monopolies initiated by government
Monopoly Creation
There are instances in which the government initiates monopolies, creating a government-granted monopoly or a government monopoly. Government-granted monopolies often closely resemble government monopolies in many respects, but the two are distinguished by the decision-making structure of the monopolist. In a government monopoly, the holder of the monopoly is formally the government itself and the group of people who make business decisions is an agency under the government’s direct authority. In a government-granted monopoly, on the other hand, the monopoly is enforced through the law, but the holder of the monopoly is formally a private firm, which makes its own business decisions.
Government-Granted Monopoly
In a government-granted monopoly, the government gives a private individual or a firm the right to be a sole provider of a good or service. Potential competitors are excluded from the market by law, regulation, or other mechanisms of government enforcement. Intellectual property rights such as copyright and patents are government-granted monopolies. Additionally, the Dutch East India Company provides a historical example of a government-granted monopoly. It was granted exclusive trading privileges with colonial possessions under mercantilist economic policy.
Government Monopoly
In a government monopoly, an agency under the direct authority of the government itself holds the monopoly, and the monopoly is sustained by the enforcement of laws and regulations that ban competition or reserve exclusive control over factors of production to the government. The state-owned petroleum companies that are common in oil-rich developing countries (such as Aramco in Saudi Arabia or PDVSA in Venezuela) are examples of government monopolies created through nationalization of resources and existing firms. The United States Postal Service is another example of a government monopoly. It was created through laws that ban potential competitors from offering certain types of services, such as first-class and standard mail delivery. Around the world, government monopolies on public utilities, telecommunications systems, and railroads have historically been common.
Postal Service : The postal service operates as a government monopoly in many countries, including the United States.
Legal Barriers
The government creates legal barriers through patents, copyrights, and granting exclusive rights to companies.
Learning objectives
- Identify the legal conditions that lead to monopolistic power.
In some cases, the government will grant a person or firm exclusive rights to produce a good or service, enabling them to monopolize the market for this good or service. Intellectual property rights, including copyright and patents, are an important example of legal barriers that give rise to monopolies.
Copyright
Copyright gives the creator of an original creative work (such as a book, song, or film) exclusive rights to it, usually for a limited time, with the intention of enabling the creator to be compensated for his or her work. The intent behind copyright is to promote the creation of new works by providing creators the opportunity to profit from their works. The copyright holder receives the right to be credited for the work, to determine who may adapt the work to other forms, who may perform the work, and who may financially benefit from it, along with other related rights. When the copyright on a work expires, the work is transferred to the public domain, enabling others to repurpose and build on the work.
Copyright : Copyright is an example of a temporary legal monopoly granted to creators of original creative works.
Patent
A patent is a limited property right the government gives inventors in exchange for their agreement to share the details of their invention with the public. During the term of the patent, the patent holder has the right to exclude others from making, using, or selling the patented invention. The patent provides incentives (1) to invent in the first place, (2) to disclose the invention once it is made, (3) to make the necessary investments in research and development, production, and bringing the invention to market, and (4) to innovate by designing around or improving upon earlier patents. When a patent expires and the invention enters the public domain, others can build on the invention.
For example, when a pharmaceutical company first markets a drug, it is usually under a patent, and only the pharmaceutical company can sell it until the patent expires. This allows the company to recoup the cost of developing this particular drug. After the patent expires, any pharmaceutical company can manufacture and sell a generic version of the drug, bringing down the price of the original drug to compete with new versions.
Government Granted Monopoly
It is also possible that there is a monopoly because the government has granted a single company exclusive or special rights. The water utility company, for example, is a monopoly in your area because it is the only organization granted the right to provide water. Another example is that the Digital Millenium Copyright Act the proprietary Macrovision copy prevention technology is required for analog video recorders. Though other forms of copy prevention aren’t prohibited, requiring Macrovision effectively gives it a monopoly and prevents more effective copy prevention methods from being developed.
Natural Monopolies
Natural monopolies occur when a single firm can serve the entire market at a lower cost than a combination of two or more firms.
Learning Obejectives
- Demonstrate an understanding of how a natural monopoly is created
Natural monopolies occur when a single firm is able to serve the entire market demand at a lower cost than any combination of two or more smaller firms. For example, imagine there are two firms in a natural monopoly’s market and each of them produces half of the quantity that the monopoly produces. The total cost of the natural monopoly is lower than the sum of the total costs of two firms producing the same quantity.
Natural Monopoly : The total cost of the natural monopoly’s production is lower than the sum of the total costs of two firms producing the same quantity.
Cost Structure
A natural monopoly’s cost structure is very different from that of most industries. In other industries, the marginal cost initially decreases due to economies of scale, then increases as the company experiences growing pains (as employees become overworked, the firm’s bureaucracy expands, etc.). Along with this, the average cost of production decreases and then increases. In contrast, a natural monopoly will have a marginal cost that is constant or declining, and an average total cost that drops as the quantity of output increases.
Fixed Costs
Natural monopolies tend to form in industries where there are high fixed costs. A firm with high fixed costs requires a large number of customers in order to have a meaningful return on investment. As it gains market share and increases its output, the fixed cost is divided among a larger number of customers. Therefore, in industries with large initial investment requirements, average total costs decline as output increases. Once a natural monopoly has been established, there will be high barriers to entry for other firms because of the large initial cost and because it would be difficult for the entrant to capture a large enough part of the market to achieve the same low costs as the monopolist.
Examples of natural monopolies are water and electricity services. For both of these, fixed costs of building the necessary infrastructure are high. The cost of constructing a competing transmission network and delivering service will be so high that it effectively bars potential competitors from entering the monopolist’s market.
Other Barriers to Entry
Firms gain monopolistic power as a result of markets’ barriers to entry, which discourage potential competitors.
Learning Obejectives
- Identify the common conditions that lead to monopolistic power
Monopolies derive their market power from barriers to entry: circumstances that prevent or greatly impede a potential competitor’s ability to compete in the market. There are several different types of barriers to entry.
Control Over Natural Resources
The supply of natural resources such as precious metals or oil deposits is limited, giving their owners monopoly powers. For example, De Beers controls the vast majority of the world’s diamond reserves, allowing only a certain number of diamonds to be mined each year and keeping the price of diamonds high.
Diamond : De Beers controls the majority of the world’s diamond reserves, preventing other players from entering the industry and setting a high price for diamonds.
High Capital Requirements
Some production processes require large investments in capital or large research and development costs that make it difficult for new companies to enter an industry. Examples include steel production, pharmaceuticals, and space transport.
Economies of Scale
Monopolies exhibit decreasing costs as output increases. Decreasing costs coupled with large initial costs give monopolies a cost advantage in production over would-be competitors. Market entrants have not yet achieved economies of scale, so their output simply costs so much more than the incumbent firms that market entry is difficult.
Network Effects
The use of a product by other people can increase its value to a person. One example is Microsoft spreadsheet and word processing software, which is still used widely. This is because when a person uses software that is used by so many others, he or she is less likely to run into compatibility problems in the course of work or other activities. This tendency to use what everyone else is using makes it difficult for new companies to develop and sell competing software.
Facebook : Network effects are one reason why it’s so difficult for new companies to compete against Facebook: they simply will have difficulty establishing a network of users to compete.
Legal Barriers
Legal rights can provide an opportunity to monopolize a market for a good. Intellectual property rights, such as patents and copyright, give the rights holder exclusive control over the production and sale of certain goods. Property rights may give a company exclusive control of the materials necessary to produce a good. The granting of permits or professional licenses can also favor certain firms, while setting standards that are difficult for new firms to meet.
Government Backing
There are cases in which a government agency is the sole provider of a particular good or service and competition is prohibited by law. For example, in many countries, the postal system is run by the government with competition forbidden by law in some or all services. Government monopolies in public utilities, telecommunications systems, and railroads have also historically been common. In other instances, the government may be an invested partner in a monopoly rather than a sole owner. This will still make it difficult for competitors to operate on equal footing.
Key Points
- Single ownership over a resource gives the owner the power to raise the market price of a good over marginal cost without losing customers to competitors.
- De Beers is a classic example of a monopoly based on a natural resource. De Beers had a lot of market power in the world market for diamonds over the course of the 20th century, keeping the price of diamonds high.
- In practice, monopolies rarely arise because of control over natural resources.
- Economies of scale are cost advantages that large firms gain because of their size.
- Natural monopolies arise as a result of economies of scale. Natural monopolies have overwhelming cost advantages over potential competitors.
- Network effects occur when the value of a good or service increases because many other people are using it. This makes competing goods or services with lower levels of adoption unattractive to new customers.
- Government-granted monopolies and government monopolies differ in the decision-making structure of the monopolist. In a government-granted monopoly, business decisions are made by a private firm. In a government monopoly, decisions are made by a government agency.
- In a government-granted monopoly, the government gives a private individual or a firm the right to be a sole provider of a good or service.
- In a government monopoly, an agency under the direct authority of the government itself holds the monopoly.
- In both types of government-initiated monopoly competition is kept out of the market through laws, regulations, and other mechanisms of government enforcement.
- Intellectual property rights are an example of legal barriers that give rise to monopolies.
- A copyright gives the creator of an original creative work exclusive rights to it for a limited time. This provides an incentive for the continued creation of innovative goods.
- A patent is a limited property right the government gives inventors in exchange for the details of their invention being made public.
- The government can provide exclusive or special rights to companies that legally allow them to be monopolies.
- A natural monopoly ‘s cost structure is very different from that of most industries. For a natural monopoly, the average total cost continues to shrink as output increases.
- Natural monopolies tend to form in industries where there are high fixed costs. A firm with high fixed costs requires a large number of customers in order to have a meaningful return on investment.
- Other firms are discouraged from entering the market because of the high initial costs and the difficulty of obtaining a large enough market share to achieve the same low costs as the monopolist.
- There are several different types of barriers to entry, including a firm ‘s control over scarce natural resources, high capital requirements for an industry, economies of scale, network effects, legal barriers, and government backing.
- Some industries require large investments in capital or research and development, making it difficult for new firms to enter.
- Monopolies benefit from economies of scale, which give them a cost advantage over their competitors.
- The legal system can grant firms monopoly rights over a resource or production of a good.
Key Terms
- market power : The ability of a firm to profitably raise the market price of a good or service over marginal cost. A firm with total market power can raise prices without losing any customers to competitors.
- economic rent : The portion of income paid to a factor of production in excess of its opportunity cost.
- economies of scale : The characteristics of a production process in which an increase in the scale of the firm causes a decrease in the long run average cost of each unit.
- Network externalities : Are evident when the value of a product or service is dependent on the number of other people using it.
- Natural monopoly : Occurs when a firm is able to serve the entire market demand at a lower cost than any combination of two or more smaller, more specialized firms.
- Government monopoly : A form of monopoly in which a government agency is the sole provider of a particular good or service and competition is prohibited by law.
- Government-granted monopoly : A form of monopoly in which a government grants exclusive rights to a private individual or firm to be the sole provider of a good or service.
- Copyright : A legal concept that gives the creator of an original work exclusive rights to it, usually for a limited time, with the intention of enabling the creator to be compensated for his or her work.
- patent : A declaration issued by a government agency declaring the inventor of a new product has the privilege of stopping others from making, using or selling the claimed invention for a limited time.
- Barriers to entry : Circumstances that prevent or greatly impede a potential competitor’s ability to compete in the market.
- Network effects : When the value of a product or service is dependent on the number of people using it.
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- BTY Chapter 15. Provided by : mrski-apecon-2008 Wikispace. Located at : https://mrski-apecon-2008.wikispaces...BTY+Chapter+15 . License : CC BY-SA: Attribution-ShareAlike
- Government-granted monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Governm...anted_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Government monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Government_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Strategy for Information Markets/Monopoly. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Strateg...oly%23Monopoly . License : CC BY-SA: Attribution-ShareAlike
- Coercive monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Coercive_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Government-granted monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Governm...ted%20monopoly . License : CC BY-SA: Attribution-ShareAlike
- Government monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Government%20monopoly . License : CC BY-SA: Attribution-ShareAlike
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- Patent. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Patent . License : CC BY-SA: Attribution-ShareAlike
- Patent. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Patent . License : CC BY-SA: Attribution-ShareAlike
- Monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Monopoly . License : CC BY-SA: Attribution-ShareAlike
- Generic drug. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Generic_drug . License : CC BY-SA: Attribution-ShareAlike
- Government-granted monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Governm...anted_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Copyright. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Copyright . License : CC BY-SA: Attribution-ShareAlike
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- Market forms. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Market_forms . License : CC BY-SA: Attribution-ShareAlike
- Strategy for Information Markets/Monopoly. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Strateg...tural_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Natural monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Natural_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Natural monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Natural_monopoly . License : CC BY-SA: Attribution-ShareAlike
- Natural monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Natural_monopoly . License : CC BY-SA: Attribution-ShareAlike
- economies of scale. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/economies%20of%20scale . License : CC BY-SA: Attribution-ShareAlike
- Natural monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Natural%20monopoly . License : CC BY-SA: Attribution-ShareAlike
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- Monopoly. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Monopoly . License : CC BY-SA: Attribution-ShareAlike
- Step 15. Monopoly- Clair. Provided by : mrski-apecon-2008 Wikispace. Located at : http://mrski-apecon-2008.wikispaces....onopoly-+Clair . License : CC BY-SA: Attribution-ShareAlike
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16.2: Chambers and Circulation through the Heart
The human heart consists of four chambers: The left side and the right side each have one atrium and one ventricle . Each of the upper chambers, the right atrium (plural = atria) and the left atrium, acts as a receiving chamber and contracts to push blood into the lower chambers, the right ventricle and the left ventricle. The ventricles serve as the primary pumping chambers of the heart, propelling blood to the lungs or to the rest of the body.
There are two distinct but linked circuits in the human circulation called the pulmonary and systemic circuits. Although both circuits transport blood and everything it carries, we can initially view the circuits from the point of view of gases. The pulmonary circuit transports blood to and from the lungs, where it picks up oxygen and delivers carbon dioxide for exhalation. The systemic circuit transports oxygenated blood to virtually all of the tissues of the body and returns relatively deoxygenated blood and carbon dioxide to the heart to be sent back to the pulmonary circulation.
The right ventricle pumps deoxygenated blood into the pulmonary trunk , which leads toward the lungs and bifurcates into the left and right pulmonary arteries . These vessels in turn branch many times before reaching the pulmonary capillaries , where gas exchange occurs: Carbon dioxide exits the blood and oxygen enters. The pulmonary trunk arteries and their branches are the only arteries in the post-natal body that carry relatively deoxygenated blood. Highly oxygenated blood returning from the pulmonary capillaries in the lungs passes through a series of vessels that join together to form the pulmonary veins —the only post-natal veins in the body that carry highly oxygenated blood. The pulmonary veins conduct blood into the left atrium, which pumps the blood into the left ventricle, which in turn pumps oxygenated blood into the aorta and on to the many branches of the systemic circuit. Eventually, these vessels will lead to the systemic capillaries, where exchange with the tissue fluid and cells of the body occurs. In this case, oxygen and nutrients exit the systemic capillaries to be used by the cells in their metabolic processes, and carbon dioxide and waste products will enter the blood.
The blood exiting the systemic capillaries is lower in oxygen concentration than when it entered. The capillaries will ultimately unite to form venules, joining to form ever-larger veins, eventually flowing into the two major systemic veins, the superior vena cava and the inferior vena cava , which return blood to the right atrium. The blood in the superior and inferior venae cavae flows into the right atrium, which pumps blood into the right ventricle. This process of blood circulation continues as long as the individual remains alive. Understanding the flow of blood through the pulmonary and systemic circuits is critical to all health professions ( Figure 16.2 ).


11.3: Merchandising Decisions
Learning OUTCOME
- Explore the techniques used to evaluate merchandising decisions.
Like all businesses, retailers must evaluate their performance on a regular basis in order to continue success, improve results, or turn-around sub-par situations. Based on the condition of the retail industry today, one might say that it is more crucial than ever to get it right. Given the different kinds of retailers it is difficult to make general statements regarding the methods used to evaluate merchandise decisions. Dick’s Sporting Goods would have a somewhat different perspective than 7-11. Macy’s would review their business differently than Big O Tires.
Despite the differences between retailers, there is one major common factor: the degree of complexity within their businesses. A review of retailer merchandise decisions reflects this complexity, revealing all of the dimensions of merchandise organization. Retail merchandise organization can be broken down from a product perspective: group, division, department, vendor, classification, sub-class and SKU. Another dimension of review could be a location perspective: region, area, district, store, department, classification and SKU. Layer on to that the time dimension: year, season, quarter, month and week. Retailers then take these various dimensions and find points of intersection depending on the level of the business undergoing analysis.
Retail merchandise analysis in usually performed both at a financial level and a unit level- again owing to the degree of complexity and level of detail needed to manage and analyze merchandise decisions.
Practice Questions
Contributors and Attributions
- Merchandising Decisions. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution


Outsourcing and Offshoring
Offshoring
means setting up operations in a low-cost country for the purpose of hiring local workers at lower labor rates. Offshoring differs from outsourcing in that the firm retains control of the operations and directly hires the employees. In
outsourcing
, by contrast, the company delegates an entire process (such as accounts payable) to the outsource vendor. The vendor takes control of the operations and runs the operations as they see fit. The company pays the outsource vendor for the end result; how the vendor achieves those end results is up to the vendor.
Companies that choose to offshore face the same location-criteria factors as companies that make production-operation decisions.
The advantages of outsourcing include the following:
-
Efficient processes (the outsourcer typically specializes in a particular process or set of processes, giving them high levels of expertise with that process)
-
Access to specialized equipment that may be too expensive for a company to invest in unless that process is their chief business
India has long been a favorite location for outsourcing services, such as call centers and software testing, because of its English-speaking, highly educated workforce. The labor-rate ratio has been five to one, meaning that a company based in the United Kingdom, for example, could hire five Indian college graduates for the price of hiring one UK college graduate. Given the high demand for their labor, however, Indian employees’ wages have begun to rise. Offshoring companies are now faced with a new challenge. The firms hire and train Indian employees only to see them leave in a year for a higher salary elsewhere. This wage inflation and high turnover in India has led some companies, like ABN AMRO Bank, to consider whether they should move offshoring operations to China, where wages are still low. The downside is that graduates in China aren’t as knowledgeable about the financial industry, and language problems may be greater.
Diageo, the world’s largest purveyor of spirits, used the following criteria when choosing an offshoring-services location. Diageo analyzed nineteen locations in fourteen countries, ultimately choosing Budapest, Hungary, as the location of its offshore shared-services operations. The primary criteria Diageo used were
-
a low-cost base, both in terms of start-up and ongoing running costs;
-
a favorable general business environment;
-
the availability of suitable staff—particularly with regard to language skills;
-
a high level of local and international accessibility with good transport links;
-
the attractiveness for international staff; and
-
a robust regulatory framework.
Sources: Burt Helm, “Diageo Targets the Home Bartender,” BusinessWeek, July 6, 2009, 48. Linda Pavey, “OTC Focus & Solutions,” June 6, 2005, accessed August 6, 2010,
http://www.ideaslab.info
.
Companies save on labor costs when offshoring, but the “hidden costs” can be significant. These hidden costs include the costs of additional facilities, telecommunications, and technological infrastructure. Delays or problems with internal project coordination and the need for redundancy can add even more costs.
Did You Know: Standard Chartered Bank Mitigated Risk by Duplicating Operations in Chennai and Kuala Lumpur
As you can imagine, banks are very concerned about security because of the highly confidential customer information they possess. Some banks try to mitigate the risks by setting up mirror sites. Standard Chartered Bank, for instance, chose Chennai in South India as the hub for its Scope International operations, but some of the tasks are also done in Kuala Lumpur in Malaysia: “Because we run the operations of 52 countries, we have to satisfy information security and business continuity issues in all locations,” says Sreeram Iyer, Group Head, Global Shared Services Centers, Standard Chartered Scope International at the time of the decision. “Kuala Lumpur backs up the Chennai center and vice versa.”
Source:


7: California’s Regions and Provinces
- Classify California's Physiographic regions.
- Understand California’s subdivision by tribal regions and political geography.
- Identify and locate the 11 geomorphic provinces in California.
California can be subdivided using several different lenses such as political, geographical, and even geological. This unit will discuss the different regions and provinces within the state, and why these boundaries exist. This unit will also set the foundation of how the next few units have subdivided certain regions of the state.
Although they share their similarities, regions and provinces are very different. A region is an area that is divided by physical, human, and environmental characteristics or boundaries. Provinces are an administrative division, meaning that boundaries are set and bound. Many of the regions in California are also provinces, and many provinces may have several regions within.
-
- 7.1: California’s Counties
- In 1850, California established its first 27 counties following recommendations from a committee led by General Mariano Vallejo, adding more and changing some names. By 1960, the number grew to 43 through further subdivisions, eventually reaching 58 counties by 1907. The state contains San Bernardino, the largest county in the contiguous U.S., and Los Angeles, the most populous. California's counties are primarily general law counties, with 14 operating under charters.
-
- 7.2: California’s Tribal Groups
- Native Californians are diverse Indigenous nations and peoples originally from the area now known as California. The state hosts 109 federally recognized tribes and over forty tribes seeking recognition, making it the state with the second-largest Native American population in the U.S.
-
- 7.3: California’s Political Geography
- California is known for its unique political culture, often seen as more liberal and a trendsetter compared to other U.S. states. It has been a stronghold for the Democratic Party, evident in its voting patterns such as the 2016 presidential election. The state has been a pioneer in numerous political initiatives including recalling governors, legalizing abortion and stem cell research, and altering election processes.
-
- 7.4: The 11 Geomorphic Provinces
- California's geomorphic provinces are distinct geologic regions, each with unique features defined by geology, faults, topographic relief, and climate. There are eleven provinces, including the Great Valley, Modoc Plateau, Klamath Mountains, Sierra Nevada, Transverse Ranges, Cascade Range, Coast Ranges, Peninsular Ranges, Basin & Range, Mojave Desert, and Colorado Desert. These provinces offer diverse landscapes.


1.4: Kinetic vs. Thermodynamic Control of Reactions
After completing this section, you should be able to
- Explain the difference between thermodynamic and kinetic control of a chemical reaction, for example, the reaction of a conjugated diene with one equivalent of hydrogen halide.
- Draw a reaction energy diagram for a reaction that can result in both a thermodynamically controlled product and a kinetically controlled product.
- Explain how reaction conditions can determine the product ratio in a reaction in which there is competition between thermodynamic and kinetic control.
Make certain that you can define, and use in context, the key terms below.
- kinetic control
- thermodynamic control
Like non-conjugated dienes, conjugated dienes are subject to attack by electrophiles. In fact, conjugated dienes experience relatively greater kinetic reactivity when reacted with electrophiles than non-conjugated dienes do. The reaction mechanism is similar to other electrophilic addition reactions to alkenes (Section 7.9) . During the electrophilic addition of HBr to a 1,3-butadiene, However, there are two possible outcomes once the carbocation intermediate is formed. The allyl carbocation is stabilized by resonance structures that vary in the position of the carbocation. This allows the bromide ion to add to either of these carbons resulting in the formation 1,2 and 1,4 addition products. When this reaction is run at a temperature of 0 o C or lower, the 1,2 addition product dominates. However, if the same reaction is run at 40 o C the 1,4 addition product dominates.
The reaction of one equivalent of hydrogen bromide with 1,3-butadiene yields different product ratios under different reaction conditions and is a classic example of the concept of thermodynamic versus kinetic control of a reaction.
At lower temperatures the formation of the 1,2 and 1,4 addition products are irreversible and thus do not reach equilibrium. When a reaction is irreversible the major product is determined by the relative reaction rates and not by thermodynamic stability. Of the two products, the formation of the 1,2 addition product has a higher rate of reaction and forms faster making it the major product. The reaction product which forms with a higher rate of reaction is called the Kinetic Product and when the kinetic product dominates, the reaction is said to be under Kinetic Control .
At higher temperatures the reaction to form both products becomes reversible and a reaction equilibrium is reached. When a reaction is reversible the major product is determined by thermodynamic stability. The 1,4 addition product is more stable and becomes the major product under these reaction conditions. The more stable product is called the Thermodynamic Product and when the thermodynamic product dominates, the reaction is said to be under Thermodynamic Control .
The 1,4 addition product is more stable because it has an internal, disubstituted double bond, and we know that as a general rule that the thermodynamic stability of an alkene increases with increasing substitution. So, when compared to the terminal, monosubstituted alkene of the 1,2 addition product, the 1,4 addition product is expected to be more stable.
A simple definition is that the kinetic product is the product that is formed faster , and the thermodynamic product is the product that is more stable . This is precisely what is happening here. The kinetic product is 3-bromobut-1-ene, and the thermodynamic product is 1-bromobut-2-ene (specifically, the trans isomer).
An explanation for the temperature influence is shown in the following energy diagram for the addition of HBr to 1,3-butadiene. The initial step in which a proton bonds to carbon #1 is the rate determining step , as indicated by the large activation energy (light gray arrow). The second faster step is the product determining step , and there are two reaction paths (colored blue for 1,2-addition and magenta for 1,4-addition).
At elevated temperatures, the products are more likely to have enough energy to overcome the reverse activation barrier for the second step allowing regeneration of the carbocation intermediate. Under these conditions, this step of the mechanism will be reversible and an equilibrium will be established. Since the system is no longer limited by temperature, the system will minimize its Gibbs free energy, which is the thermodynamic criterion for chemical equilibrium. This places the reaction under thermodynamic control and the most thermodynamically stable molecule, the 1,4 addition product, will be predominantly formed.
If the reaction temperature is kept sufficiently low , the products will not have enough energy to overcome the reverse activation barrier to regenerate the carbocation intermediate, making this step of the mechanism effectively irreversible. This reaction is under kinetic control meaning the product which forms faster, the kinetic product, will predominate.
Of the two reaction pathways, the 1,2-addition has a smaller activation energy and would be expected to have a higher reaction rate than the 1,4-addition. The high reaction rate for 1,2-addition can be attributed to the formation of an ion pair during the reaction mechanism. This means that, after a double bond is protonated, the halide counterion remains in close proximity to the carbocation generated. Immediately following the dissociation of HX, the chloride ion is going to be much closer to C−2 than it is to C−4, and therefore attack at C−2 is much faster. This ion pair mechanism is a pre-exponential constant effects that is attributed to the proximity and frequency of collision rather than a activation barrier effect.
Experimental Evidence for the Ion-Pair Mechanism
In 1979, Nordlander et al. carried out a similar investigation on the addition of DCl (deuterated hydrochloric acid) to a different substrate, 1,3-pentadiene. This experiment was ingenious, because it was designed to proceed via an almost symmetrical intermediate:
Resonance forms 7a and 7b are both allylic and secondary. There is a very minor difference in their stabilities arising from the different hyperconjugative ability of C-D vs C-H bonds, but in any case, it is not very large. Therefore, if we adopt the explanation in the previous section, one would expect there not to be any major kinetic pathway, and both 1,2- and 1,4-addition products ( 8 and 9 ) would theoretically be formed roughly equally.
Instead, it was found that the 1,2-addition product was favored over the 1,4-addition product. For example, at -78 o C in the absence of solvent, there was a roughly (75:25) ratio of 1,2- to 1,4-addition products. Clearly, there is a factor that favors 1,2-addition that does not depend on the electrophilicity of the carbon being attacked! The authors attributed this effect to an ion pair mechanism. This means that, after the double bond is protonated (deuterated in this case), the chloride counterion remains in close proximity to the carbocation generated. Immediately following dissociation of DCl, the chloride ion is going to be much closer to C-2 than it is to C-4 , and therefore attack at C -2 is much faster. In fact, normal electrophilic addition of HX to conjugated alkenes in polar solvents can also proceed via similar ion pair mechanisms. This is reflected by the greater proportion of syn addition products to such substrates.
The reaction of one equivalent of hydrogen bromide with 1,3-butadiene yields different products. Why is the 1,4-addition product the thermodynamically more stable product?
- Answer
-
The 1,4- product is more thermodynamically stable because there are two alkyl groups on each side of the double bond, and more substituted alkenes are more stable.
The Addition of 1 equivalent of bromine to 2,4-hexadiene at 0 degrees C gives 4,5-dibromo-2-hexene plus an isomer. What is the structure of that isomer?
- Answer
-
Your time to draw the structure of 2,5-Dibromo-3-hexene
For the following reaction:
a- Identify the kinetically controlled product.
b- which product is the result of 1,4-addition?
- Answer
-
a- 3-Chloro-1-butene b- 1-Chloro-2-butene
What would be the major product of the addition of HBr to 2,3-dimethyl-1,3-cyclohexadiene under thermodynamic conditions?
- Answer
-
Consider the reaction with 1,3-buta-diene reacting with HCl. Propose a mechanism for the reaction. Also, Predict why the 1,4 adduct is the major product in this reaction compared to the 1,2.
- Answer
-
Even though the cation would prefer to be in a secondary position in the transition state, the final product is less stable with a terminal alkene. Therefore the major product will be the 1,4 adduct.


7.6: Summary
The topics covered in this chapter can be summarized as follows:
| Section | Summary |
|---|---|
| 7.1 Controls Over Metamorphic Processes | Metamorphism is controlled by five main factors: the composition of the parent rock, the temperature to which the rock is heated, the amount and type of pressure, the volumes and compositions of aqueous fluids that are present, and the amount of time available for metamorphic reactions to take place. |
| 7.2 Classification of Metamorphic Rocks | Metamorphic rocks are classified on the basis of texture and mineral composition. Foliation is a key feature of metamorphic rocks formed under directed pressure. Foliated metamorphic rocks include slate, phyllite, schist, and gneiss. Metamorphic rocks formed in environments without strong directed pressure include hornfels, marble, and quartzite, although the latter two may form in high-pressure situations but not develop obvious foliated textures. |
| 7.3 Plate Tectonics and Metamorphism | Almost all metamorphism can be explained by plate-tectonic processes. Oceanic crustal rock can be metamorphosed near the spreading ridge where it was formed, but most other regional metamorphism takes place in areas where mountain ranges have been created, which are most common at convergent boundaries. Contact metamorphism takes place around magma bodies in the upper part of the crust, which are also most common above convergent boundaries. |
| 7.4 Regional Metamorphism | Geologists classify metamorphic rocks based on some key minerals—such as chlorite, garnet, andalusite, and sillimanite—that form at specific temperatures and pressures. Most regional metamorphism takes place beneath mountain ranges because the crust becomes thickened and rocks are pushed down to great depths because of the isostatic relationship between the crust and mantle. When mountains erode, those metamorphic rocks are uplifted by crustal rebound. |
| 7.5 Contact Metamorphism and Hydrothermal Processes | Contact metamorphism takes place around magma bodies that have intruded into cool rocks at high levels in the crust. Heat from the magma is transferred to the surrounding country rock, resulting in mineralogical and textural changes. Water from a cooling body of magma, or from convection of groundwater produced by the heat of the pluton, can also lead to metasomatism, hydrothermal alteration, and accumulation of valuable minerals in the surrounding rocks. |
Answers to Review Questions can be found in Appendix 2 .
- What are the two main agents of metamorphism, and what are their respective roles in producing metamorphic rocks?
- Into what metamorphic rocks will a mudrock be transformed at very low, low, medium, and high metamorphic grades?
- Why doesn’t granite change very much at lower metamorphic grades?
- Describe the main process of foliation development in a metamorphic rock such as schist.
- What process contributes to metamorphism of oceanic crust at a spreading ridge?
- How do variations in the geothermal gradient affect the depth at which different metamorphic rocks form?
- Blueschist metamorphism takes place within subduction zones. What are the unique temperature and pressure characteristics of this geological setting?
- Rearrange the following minerals in order of increasing metamorphic grade: biotite, garnet, sillimanite, chlorite.
- Why does contact metamorphism not normally take place at significant depth in the crust?
- What is the role of magmatic fluids in metamorphism that takes place adjacent to a pluton?
- How does metasomatism differ from regional metamorphism?
- How does the presence of a hot pluton contribute to the circulation of groundwater that facilitates metasomatism and hydrothermal processes?
- What must be present in the country rock to produce a skarn?
-
Two things that a geologist first considers when looking at a metamorphic rock are what the parent rock might have been, and what type of metamorphism has taken place. This can be difficult to do, even if you have the actual rock in your hand, but give it a try for the following metamorphic rocks:
- Chlorite schist
- Slate
- Mica-garnet schist
- Amphibolite
- Marble


8.10: Introduction to Common Group Behaviors
What you’ll learn to do: describe common group behaviors that can help or hurt organization goals
People are social animals and function best when they work with others in groups or teams. Some human behaviors show up repeatedly among groups. In the workplace, it is important to understand how these behaviors affect performance. Social cohesion and collective efficacy benefit team functions, whereas social loafing reduces team effectiveness. This section will examine these common behaviors.
Contributors and Attributions
CC licensed content, Original
- Introduction to Common Group Behaviors. Authored by : John/Lynn Bruton and Lumen Learning. License : CC BY: Attribution